Voltammetry based on fractional diffusion.

A cyclic voltammetric experiment governed by anomalous diffusion of an electroactive species is theoretically analyzed by means of fractional calculus. The diffusion mass transfer under semi-infinite conditions at a planar electrode is ascribed by a Fick's second law-like differential equation in which the time derivative of the concentration function is of a fractional order alpha, ranging from 0 to 1. Rigorous solutions relating the concentrations of electroactive species with the electric current are derived by means of the Wright function for the case of a simple reversible electrode reaction of two chemically stable redox-active species having identical diffusion coefficients. A general mathematical solution for a voltammetric experiment, relating the surface concentrations with the current and electrode potential, is presented in the form of an integral equation. On the basis of the latter solution, the cyclic voltammetric experiment is simulated under variety of conditions, in order to inspect the influence of the fractional parameter alpha and to reveal its physical significance. Aiming to explain peculiar features of cyclic voltammograms, concentration profiles of electroactive species, together with the Cottrell-like equation, are analyzed for various alpha values. It has been established that the shape of a cyclic voltammogram depends strongly on alpha, varying from a steady-state sigmoid shape when alpha --> 0 to a conventional peak-like shape for alpha --> 1. Whereas the midpeak potential is independent of alpha, the peak currents are proportional to upsilon(alpha/2), where upsilon is the sweep rate.